Fuzzy histograms and fuzzy probability distributions

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1 Fuzzy histograms and fuzzy probability distributions R. Viertl Wiedner Hauptstraße 8 / 7-4 Vienna R.Viertl@tuwien.ac.at W. Trutschnig Wiedner Hauptstraße 8 / 7-4 Vienna trutschnig@statistik.tuwien.ac.at bstract Given fuzzy data the concept of histograms has to be etended to so-called fuzzy histograms in order to incorporate fuzziness into considerations. For these fuzzy histograms the height over a class itself is fuzzy. Based on this the etension of classical probability distributions to so-called fuzzy probability distributions on arbitrary σ-algebras is described, the main properties of fuzzy probability distributions are analyzed, and the characteristics of fuzzy probability distributions are eplained. Keywords: Fuzzy data, Imprecise probability Notation used The following notation will be used throughout the paper: characterizing function ξ ( ) of a fuzzy number is a real function with the following properties:. ξ () R 2. α (,] the set [ξ ] α = [ ] α := { R : ξ () α}, the so called α-cut, is a non-empty compact interval. The set of all fuzzy numbers will be denoted by F. λ denotes the Lebesgue measure on R. If [a,b ] and [a 2,b 2 ] are compact intervals then [a,b ] [a 2,b 2 ] is defined by: [a,b ] [a 2,b 2 ] : a a 2 and b b 2 For fuzzy numbers,y F, y is defined by: y : [ ] α [y ] α α (,] () For fuzzy numbers,y F, y is defined by: y : [] α [y ] α α (,] (2) For fuzzy numbers,y F, y denotes the common sum and y denotes the common difference of fuzzy numbers, i.e. the α-cut of y ( y ) is the Minkowski sum (difference) of the α-cuts of and y (compare [3]). 2 Fuzzy histograms Given a fuzzy sample,, n and a crisp partition K,,K m of the reals R, the first problem that arises form the statistical point of view is, how the concept of relative frequencies and histograms can be etended naturally from the idealized case of real-valued samples. In this respect the most important thing that has to be considered is, that it may happen, that an element i is not contained in a single

2 class but partially lies in different classes as depicted in Figure (ξ i ( ) as common denotes the characterizing function of i ). i () that h n,α (K) decreases if α increases. Consequently ( [h n,α (K),h n,α (K)] ) is a family α (,] of compact non-empty intervals for n and K fied, that decreases if α increases, i.e. [ hn,α (K),h n,α (K) ] [ h n,β (K),h n,β (K) ] a holds for α β and α,β (,]. It can be seen easily that unfortunately in general ( [h n,α (K),h n,α (K)] ) is not α (,] a family of α-cuts of a fuzzy number. However the conve hull of the family ( [h n,α (K),h n,α (K)] ) denoted by α (,] h n(k) F, which is a fuzzy number, can be built. It is easy to see that h n(k) F fulfills K i K i+ Figure : Fuzzy observation i with classification problem In order to get a grip on this classification problem one can proceed as follows: For every α (,] and every set K R define the lower relative frequency of level α, denoted by h n,α (K), and the upper relative frequency of level α, denoted by h n,α (K), by h n,α (K):= #{ i : [ i ] α K } (3) n and h n,α (K):= #{ i : [ i ] α K } (4) n respectively. Thus the lower relative frequency of level α counts all i {,2,,n} for which the α-cut of i is contained in the set K and divides by n, whereas the upper relative frequency of level α counts all i {,2,,n} for which the α-cut of i has non-empty intersection with the set K and divides by n. Since obviously h n,α (K) h n,α (K) holds for every n N, for every α (,] and every K R, it follows immediately that ( [hn,α (K),h n,α (K)] ) α (,] is a family of compact non-empty intervals (for n and K fied) in α. Furthermore again for n and K fied it follows immediately from the definition that h n,α (K) increases if α increases and [h n (K)] α = [ h n,α (K),h n,α (K) ] (5) for λ-almost every α (,] (compare for instance [6]). Figure 2 and Figure 3 depict a fuzzy sample of size and the corresponding fuzzy relative frequency h ([,2]) of the set [,2] respectively. i () a 2 h n ([,2]) a Figure 2: Fuzzy sample of size Figure 3: Fuzzy relative frequency h ([,2])

3 n aonometric sketch of a fuzzy histogram is depicted in Figure 4. more general concept of so-called fuzzy probability distributions induced by fuzzy probability densities can be developed. Remarkably this concept fulfills all the properties listed in Lemma 2. in a very general setting, which is a strong argument for defining general fuzzy probability distributions in the way it is done at the end of the paper. 3 Fuzzy probability densities Figure 4: Fuzzy histogram s a fuzzy-valued mapping h n has the following properties, which are easy to verify: Lemma 2. Suppose that, 2,..., n is a fuzzy sample of size n and that B,C are arbitrary subsets of R, then:. supp ( h n(b) ) [,] 2. h n (R) = {}, h n ( ) = {} 3. B C R h n(b) h n(c) 4. B C= h n(b C) h n(b) h n(c) 5. h n (Bc ) = {} h n (B) Having in mind both the Strong Law of Large Numbers (compare []) and the fact that h n(b) is a fuzzy number for every set B R it is inevitable to consider fuzzy-valued probabilities as generalizations of classical probabilities. Of course the question immediately arises, which properties a fuzzy-valued mapping P : F on a σ-algebra should fulfill in order to be called probability, however Lemma 2. suggests what properties a meaningful notion at least must satisfy. Such a notion will be called fuzzy probability distribution. In the sequel a natural approach to such fuzzy probability distributions based on a completely different idea will be discussed. In the elementary case, this approach goes back to Buckley s fuzzy probabilities ([2]), however filtering out the main idea, a much If f : Ω F is a fuzzy-valued function on an arbitrary measure space (Ω,,µ), then the function F α, defined by F α (t) := [f (t)] α, (6) is an interval-valued function on Ω for each α (,]. The lower and upper α-level functions f α ( ) and f α ( ) are defined by f α (t) := min ( F α (t) ) (7) f α (t) := ma ( F α (t) ) (8) for every t Ω and every α (,]. function f is called selection of the intervalvalued function F α, if f(t) F α (t) holds for µ-almost every t Ω. For every α (,] the set of all measurable selections of F α will be denoted by Sel(F α ). Definition 3. fuzzy-valued function f on a measure space (Ω,,µ) is called (uniformly) integrably bounded, if there eists an integrable function h such that y h(t) for every y supp(f (t)) holds for µ-almost every t Ω. Using these notions the definition of a fuzzy probability density can be stated as follows: Definition 3.2 Let (Ω,,µ) be an arbitrary measure space. Then a function f : Ω F is called fuzzy probability density with respect to the measure µ on (Ω, ), if for F α the following two conditions are fulfilled:. F α (t) [, ) t Ω, α (,] 2. f Sel(F ) such that Ω f(t)dµ(t) =

4 Definition 3.2 includes and unifies z discrete fuzzy probabilities in Buckley s sense (the measure in this case is the counting measure) classical probability densities on a measure space (Ω,,µ) classical parametric probability densities (depending continuously on the parameter) with a fuzzy parameter as treated in [2] (compare Eample 3.3) t Eample 3.3 Consider the well-known eponential density g η with parameter η, defined by g η (t) = η e ηt for every t and η >. Obviously for every t [, ) the function h t : (, ) (, ), defined by h t (η) := g η (t) = η e ηt, is a continuous (in fact even C -) function in η (, ). Therefore if the paramter η is no real number but a fuzzy number η F with support in (, ) then applying the wellknown Etension Principle to the function h t, f (t), defined by f (t) := h t (η ) is a fuzzy number for every t [, ). Consequently f ( ) is a fuzzy-valued mapping on [, ). Moreover because of the fact that for every η [η ] the function g η ( ) is both a measurable selection of F ( ), and a probability density on R with respect to the Lebesgue measure λ it follows that f is a fuzzy probability density on the measure space (R +, B(R+ ),λ) according to Definition 3.2. fuzzy probability density of eponential type may appear as depicted in Figures 5 and 6 (produced with MTLB). 4 Fuzzy probability distributions induced by fuzzy probability densities Given a fuzzy probability density f with respect to the measure µ on (Ω, ), a fuzzy Figure 5: Sketch of a fuzzy probability density.5 z Figure 6: Sketch from another focus probability distribution P on can be defined (analogous the discrete case treated by Buckley) on all sets by { [P ()] α := f(t)dµ(t) : f D α }, (9) whereby { } D α := f Sel(F α ) : f(t)dµ(t) =. Ω ( () Using the notation used in [9] and [] equation (9) can be also be epressed as P () = = f (t)dµ(t). ) In other words, [P ()] α is the set of all probabilities of induced by all classical probability densities that are selections of the t

5 interval-valued mapping F α. The ( following theorem implies that [P ) ()] α really is the family of α (,] α-cuts of a fuzzy number: Theorem 4. Let (Ω,, µ) be a measure space, a measurable set, f : Ω F a fuzzy probability density with respect to the measure µ, f integrably bounded, and [P ()] α defined according to (9), then:. [P ()] α is a non-empty compact interval for every α (,] 2. [P ()] α is a nested family monotonically decreasing in α with [P ()] α = [P ()] β. α<β Sketch of the proof: If a function f is a measurable selection of F α and g is another measurable function such that f(t) = g(t) holds for µ-almost every t Ω, then obviously g is a selection of F α too. Consequently the family D α in (9) can be replaced by the family D α, defined by { D α := f L (Ω,,µ) : f Sel(F α ) and } f(t)dµ(t) =. () Ω pplying results from functional analysis (compare [4]), that characterize weakly compact subsets of L (Ω,,µ), it can be shown that - under the assumptions of Theorem 4. - the family D α is a weakly compact (and conve) subset of L (Ω,,µ). The mapping Φ, defined by Φ : L (Ω,,µ) f R f(t)dµ(t), is a continuous linear functional on L (Ω,,µ) and therefore also continuous in the weak topology. Moreover it satisfies Φ (D α ) = [P ()] α α (,]. Φ preserves compactness and conveity, which proves the compactness and conveity of [P ()] α. part from that [P ()] α holds for every α (,] by definition, which proves that [P ()] α is a non-empty compact interval for every α (,]. The second assertion of the theorem can be proved similarly. t first sight it may seem laborious to calculate the fuzzy number P () F in concrete eamples - however assuming some measurability conditions it turns out to be quite easy in many cases. For a precise formulation compare [5]. Eample 4.2 Eponential density with fuzzy parameter η = [,2] : In this case it is very easy to calculate the lower and upper α-level functions f α and f α of f eplicitly by elementary calculus, which yields the following result: f (t) = f (t) = { e t if t [,ln 2] 2e 2t if t (ln 2, ) 2e 2t if t [,/2] t e if t (/2,] e t if t (, ) } Having this it is for eample straigthforward to show that P ([,2]) = [ e 2, e 4 ] P ([,/2]) = [ e /2, e ]. and Eample 4.3 Eponential density with fuzzy parameter η = τ F defined by: τ () := 2 2 if [, 3 2 ] if ( 3 2,2] otherwise R. For this case too it is possible to calculate the lower and upper boundary functions f α and f α as well as P ([,2]) and P ([,/2]) eplicitly by elementary calculus, which yields [ P ([,2]) ] α = [ e 2 α, e 4+α] [ P ([,/2]) ] α = [ e 2 α 4, e + α 4 ]

6 for every α (,]. In Figure 4 P ([,/2]) and P ([,2]) for the case of η = τ are depicted by solid lines. Furthermore for comparison the corresponding results for the case η = [,2] (Eample 4.2) are depicted by dotted lines. P ([,/2]) P ([,2]) e -/2 -e - -e -2 -e -4 Figure 7: P ([,/2]) and P ([,2]) for the eponential density with fuzzy parameter η = τ and η = [,2] (dotted) It will be shown now that P satisfies the same assertions as stated in Lemma 2. (amongst others): Theorem 4.4 Let (Ω,, µ) be an arbitrary measure space, f : Ω F a fuzzy probability density with respect to the measure µ, f integrably bounded,,b measurable sets, and P defined according to (9), then:. supp(p ()) [,] 2. P (Ω) = {}, P ( ) = {} 3. B = P () P (B) (Monotony) 4. B = = P ( B) P () P (B) (Subadditivity) 5. P ( c ) = {} P () Proof: Points one and two are clear by definition. The third point immediately follows from the fact that for B f(t)dµ(t) f(t)dµ(t) f D α. B Suppose now that B = and that [P ( B)] α. Then there eists a function f D α such that = B f(t)dµ(t) and therefore = f(t)dµ(t) = B = f(t)dµ(t) + f(t)dµ(t) B [P ()] α [P (B)] α. In order to prove the fifth point suppose now that [P ( c )] α. Then there eists a function f D α with = f(t)dµ(t) = c = f(t)dµ(t) f(t)dµ(t) = Ω = f(t)dµ(t) [P ()] α. On the other hand if [P ()] α, then there eist y [P ()] α and f D α such that = y = f(t)dµ(t) = = f(t)dµ(t) f(t)dµ(t) = Ω = f(t)dµ(t) [P ( c )] α. Ω\ 5 General fuzzy probability distributions Looking back at the properties of the fuzzy relative frequencies stated in Lemma 2. and that of the fuzzy probability distributions induced by integrably bounded fuzzy probability densities as formulated in Theorem 4.4, then the following definition of a general fuzzy probability distribution is suggestive: Definition 5. Suppose that is a σ-field in Ω, then a fuzzy-valued function P : F is called (general) fuzzy probability distribution on Ω, if the following four conditions are fulfilled:. P (Ω) = {}, P ( ) = {}

7 2. If,B, B, then P () P (B) holds. 3. If,B, B =, then P ( B) P () P (B) holds. 4. If, then P ( c ) = {} P () holds. Considering [ ] the α-level functions p α ( ),p α ( ) of P : F, defined by [ ] p α (),p α () := [P ()] α, (2) whereby α (,] and, Definition 5. can be reformulated equivalently as follows: Definition 5.2 Suppose that is a σ-field in Ω, then a fuzzy-valued function P : F is called (general) fuzzy probability[ distribution] on Ω, if the α-level functions p α ( ),p α ( ) defined according to (2) fulfill the following four conditions:. p α (Ω)=p α (Ω)=, p α ( )=p α ( )= α (,] 2. If,B, B, then p α () p α (B) and p α () p α (B) holds for all α (,]. 3. For every α (,] p α ( ) is superadditive and p α ( ) is subadditive, i.e. if,b and B =, then p α ( B) p α () + p α (B) and p α ( B) p α () + p α (B) holds. 4. For every and every α (,] the identities p α ( c ) = p α () and p α ( c ) = p α () hold. Remark: It is clear that every probability measure P on (Ω, ) can be seen as a fuzzy probability distribution by simply defining P () := {P()} F. Final Remark: It can be shown that fuzzy probability distributions are also induced by fuzzy random variables X : Ω F on an arbitrary probability space (Ω,, P) in a very natural way (for a definition of fuzzy random variables compare for instance [7]). In fact for every α (,] and every ω Ω set X α (ω) := [X (ω)] α, and define for every Borel set B B(R) (analogous to fuzzy relative frequencies) π α (B) := P ( {ω Ω : X α (ω) B} ) (3) π α (B) := P ( {ω Ω : X α (ω) B } ). It can be shown that ( [π α (B),π α (B)] ) α (,] is a a nested, monontonically decrasing family of non-empty compact intervals in α (but unfortunately in general not a family of α-cuts of a fuzzy number). Consequently building the conve hull of the family ( [π α (B),π α (B)] ) denoted by α (,] P (B) for every Borel set B a mapping P : B(R) F is defined. It can be shown that this mapping fulfills Definition 5. too. In other words, the distribution of a fuzzy random variable induces a fuzzy probability distribution in the sense of Definition 5.. This is a further justification for the general definition of a fuzzy probability distribution. References [] H. Bauer: Wahrscheinlichkeitstheorie, W. de Gruyter Verlag, Berlin New York, 22 [2] J.J. Buckley: Fuzzy Probabilities, Physica, Heidelberg New York, 23 [3] P. Diamond, P. Kloeden: Metric Spaces of Fuzzy Sets, World Scientific, Singapore, 994 [4] N. Dunford, J.T. Schwartz: Linear Operators, Wiley-Interscience, New York, 958 [5] D. Hareter, R. Viertl: Fuzzy Information and Bayesian Statistics, in M. Lopez- Diaz, M.. Gil, P. Grzegorzewski, O. Hryniewicz, J. Lawry (Eds.): Soft Methodology and Random Information Systems, Springer-Verlag, Heidelberg, pp (24)

8 [6] V. Krätschmer: Some complete metrics on spaces of fuzzy subsets, Fuzzy sets and systems 3, (22) [7] M.L. Puri, D.. Ralescu: Fuzzy random variables, J. Math. nal. ppl. 4, (986) [8] W. Trutschnig, D. Hareter: Fuzzy Probability Distributions, in M. Lopez- Diaz, M.. Gil, P. Grzegorzewski, O. Hryniewicz, J. Lawry (Eds.): Soft Methodology and Random Information Systems, Springer-Verlag, Heidelberg, pp (24) [9] R. Viertl, D. Hareter: Generalized Bayes theorem for non-precise a-priori distribution, Metrika 59, (24) [] R. Viertl, D. Hareter: Fuzzy information and imprecise probability, Zeitschrift für ngewandte Mathematik und Mechanik 84, (24) [] G. Wang, Y. Zhang: The theory of fuzzy stochastic processes, Fuzzy sets and systems 5, 6-78 (992)